Properties

Label 3420.2.bj.c.1189.6
Level $3420$
Weight $2$
Character 3420.1189
Analytic conductor $27.309$
Analytic rank $0$
Dimension $20$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3420,2,Mod(1189,3420)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3420, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3420.1189");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3420 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3420.bj (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(27.3088374913\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 20 x^{18} + 261 x^{16} - 1994 x^{14} + 11074 x^{12} - 39211 x^{10} + 99376 x^{8} - 134299 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 380)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1189.6
Root \(1.74361 + 1.00667i\) of defining polynomial
Character \(\chi\) \(=\) 3420.1189
Dual form 3420.2.bj.c.2629.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.0408382 - 2.23570i) q^{5} -1.34403i q^{7} -5.25594 q^{11} +(-2.10918 + 1.21773i) q^{13} +(-1.17765 - 0.679914i) q^{17} +(2.89815 + 3.25587i) q^{19} +(7.05514 - 4.07329i) q^{23} +(-4.99666 + 0.182603i) q^{25} +(1.03597 + 1.79435i) q^{29} -0.513207 q^{31} +(-3.00483 + 0.0548876i) q^{35} -5.57175i q^{37} +(-2.70353 + 4.68265i) q^{41} +(-11.0197 - 6.36221i) q^{43} +(-2.82785 + 1.63266i) q^{47} +5.19359 q^{49} +(-10.1892 + 5.88276i) q^{53} +(0.214643 + 11.7507i) q^{55} +(-0.0175979 + 0.0304805i) q^{59} +(0.518372 + 0.897846i) q^{61} +(2.80861 + 4.66574i) q^{65} +(0.664028 - 0.383377i) q^{67} +(-5.68450 + 9.84583i) q^{71} +(-1.86429 - 1.07635i) q^{73} +7.06413i q^{77} +(-6.48576 + 11.2337i) q^{79} +4.20304i q^{83} +(-1.47199 + 2.66062i) q^{85} +(3.65426 + 6.32937i) q^{89} +(1.63667 + 2.83479i) q^{91} +(7.16079 - 6.61235i) q^{95} +(0.721716 + 0.416683i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - q^{5} + 14 q^{19} + 9 q^{25} + 16 q^{29} + 8 q^{31} + 2 q^{35} - 26 q^{41} - 44 q^{49} - 12 q^{55} - 4 q^{59} + 2 q^{61} + 18 q^{65} + 2 q^{71} - 16 q^{79} - 39 q^{85} + 40 q^{89} - 4 q^{91} + 43 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3420\mathbb{Z}\right)^\times\).

\(n\) \(1711\) \(1901\) \(2737\) \(3061\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.0408382 2.23570i −0.0182634 0.999833i
\(6\) 0 0
\(7\) 1.34403i 0.507994i −0.967205 0.253997i \(-0.918255\pi\)
0.967205 0.253997i \(-0.0817454\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.25594 −1.58473 −0.792363 0.610050i \(-0.791149\pi\)
−0.792363 + 0.610050i \(0.791149\pi\)
\(12\) 0 0
\(13\) −2.10918 + 1.21773i −0.584980 + 0.337738i −0.763110 0.646269i \(-0.776329\pi\)
0.178130 + 0.984007i \(0.442995\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.17765 0.679914i −0.285621 0.164903i 0.350344 0.936621i \(-0.386065\pi\)
−0.635965 + 0.771718i \(0.719398\pi\)
\(18\) 0 0
\(19\) 2.89815 + 3.25587i 0.664882 + 0.746949i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.05514 4.07329i 1.47110 0.849339i 0.471625 0.881799i \(-0.343668\pi\)
0.999473 + 0.0324603i \(0.0103342\pi\)
\(24\) 0 0
\(25\) −4.99666 + 0.182603i −0.999333 + 0.0365207i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.03597 + 1.79435i 0.192375 + 0.333203i 0.946037 0.324059i \(-0.105048\pi\)
−0.753662 + 0.657262i \(0.771714\pi\)
\(30\) 0 0
\(31\) −0.513207 −0.0921747 −0.0460873 0.998937i \(-0.514675\pi\)
−0.0460873 + 0.998937i \(0.514675\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.00483 + 0.0548876i −0.507910 + 0.00927770i
\(36\) 0 0
\(37\) 5.57175i 0.915991i −0.888955 0.457995i \(-0.848568\pi\)
0.888955 0.457995i \(-0.151432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.70353 + 4.68265i −0.422220 + 0.731307i −0.996156 0.0875933i \(-0.972082\pi\)
0.573936 + 0.818900i \(0.305416\pi\)
\(42\) 0 0
\(43\) −11.0197 6.36221i −1.68048 0.970228i −0.961339 0.275369i \(-0.911200\pi\)
−0.719146 0.694859i \(-0.755467\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.82785 + 1.63266i −0.412485 + 0.238148i −0.691857 0.722035i \(-0.743207\pi\)
0.279372 + 0.960183i \(0.409874\pi\)
\(48\) 0 0
\(49\) 5.19359 0.741942
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.1892 + 5.88276i −1.39960 + 0.808060i −0.994351 0.106146i \(-0.966149\pi\)
−0.405250 + 0.914206i \(0.632815\pi\)
\(54\) 0 0
\(55\) 0.214643 + 11.7507i 0.0289425 + 1.58446i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.0175979 + 0.0304805i −0.00229105 + 0.00396822i −0.867169 0.498015i \(-0.834063\pi\)
0.864878 + 0.501983i \(0.167396\pi\)
\(60\) 0 0
\(61\) 0.518372 + 0.897846i 0.0663707 + 0.114957i 0.897301 0.441419i \(-0.145525\pi\)
−0.830930 + 0.556376i \(0.812191\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.80861 + 4.66574i 0.348366 + 0.578714i
\(66\) 0 0
\(67\) 0.664028 0.383377i 0.0811239 0.0468369i −0.458889 0.888493i \(-0.651753\pi\)
0.540013 + 0.841657i \(0.318419\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −5.68450 + 9.84583i −0.674625 + 1.16849i 0.301953 + 0.953323i \(0.402362\pi\)
−0.976578 + 0.215163i \(0.930972\pi\)
\(72\) 0 0
\(73\) −1.86429 1.07635i −0.218199 0.125977i 0.386917 0.922115i \(-0.373540\pi\)
−0.605116 + 0.796137i \(0.706873\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 7.06413i 0.805032i
\(78\) 0 0
\(79\) −6.48576 + 11.2337i −0.729705 + 1.26389i 0.227302 + 0.973824i \(0.427009\pi\)
−0.957008 + 0.290062i \(0.906324\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.20304i 0.461343i 0.973032 + 0.230672i \(0.0740923\pi\)
−0.973032 + 0.230672i \(0.925908\pi\)
\(84\) 0 0
\(85\) −1.47199 + 2.66062i −0.159660 + 0.288585i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.65426 + 6.32937i 0.387351 + 0.670912i 0.992092 0.125510i \(-0.0400568\pi\)
−0.604741 + 0.796422i \(0.706723\pi\)
\(90\) 0 0
\(91\) 1.63667 + 2.83479i 0.171569 + 0.297166i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 7.16079 6.61235i 0.734681 0.678413i
\(96\) 0 0
\(97\) 0.721716 + 0.416683i 0.0732791 + 0.0423077i 0.536192 0.844096i \(-0.319862\pi\)
−0.462913 + 0.886404i \(0.653196\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.40992 + 12.8344i 0.737315 + 1.27707i 0.953700 + 0.300759i \(0.0972400\pi\)
−0.216385 + 0.976308i \(0.569427\pi\)
\(102\) 0 0
\(103\) 9.40773i 0.926971i 0.886104 + 0.463486i \(0.153401\pi\)
−0.886104 + 0.463486i \(0.846599\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 12.8130i 1.23868i −0.785124 0.619338i \(-0.787401\pi\)
0.785124 0.619338i \(-0.212599\pi\)
\(108\) 0 0
\(109\) 0.996875 1.72664i 0.0954833 0.165382i −0.814327 0.580406i \(-0.802894\pi\)
0.909810 + 0.415025i \(0.136227\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 8.34647i 0.785170i 0.919716 + 0.392585i \(0.128419\pi\)
−0.919716 + 0.392585i \(0.871581\pi\)
\(114\) 0 0
\(115\) −9.39474 15.6068i −0.876064 1.45534i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −0.913823 + 1.58279i −0.0837700 + 0.145094i
\(120\) 0 0
\(121\) 16.6249 1.51136
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.612300 + 11.1636i 0.0547658 + 0.998499i
\(126\) 0 0
\(127\) −17.8240 + 10.2907i −1.58163 + 0.913153i −0.587005 + 0.809583i \(0.699693\pi\)
−0.994622 + 0.103569i \(0.966974\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 5.36554 9.29339i 0.468790 0.811967i −0.530574 0.847639i \(-0.678024\pi\)
0.999364 + 0.0356712i \(0.0113569\pi\)
\(132\) 0 0
\(133\) 4.37598 3.89519i 0.379446 0.337756i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 14.8771 8.58931i 1.27104 0.733834i 0.295855 0.955233i \(-0.404396\pi\)
0.975183 + 0.221399i \(0.0710622\pi\)
\(138\) 0 0
\(139\) 3.66394 + 6.34613i 0.310771 + 0.538272i 0.978530 0.206106i \(-0.0660793\pi\)
−0.667758 + 0.744378i \(0.732746\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.0857 6.40033i 0.927033 0.535223i
\(144\) 0 0
\(145\) 3.96932 2.38939i 0.329634 0.198428i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.12292 10.6052i 0.501609 0.868812i −0.498389 0.866953i \(-0.666075\pi\)
0.999998 0.00185904i \(-0.000591751\pi\)
\(150\) 0 0
\(151\) −11.5577 −0.940549 −0.470274 0.882520i \(-0.655845\pi\)
−0.470274 + 0.882520i \(0.655845\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.0209584 + 1.14737i 0.00168342 + 0.0921593i
\(156\) 0 0
\(157\) 3.58528 + 2.06996i 0.286137 + 0.165201i 0.636198 0.771526i \(-0.280506\pi\)
−0.350062 + 0.936727i \(0.613839\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.47460 9.48229i −0.431459 0.747309i
\(162\) 0 0
\(163\) 9.41672i 0.737575i 0.929514 + 0.368787i \(0.120227\pi\)
−0.929514 + 0.368787i \(0.879773\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.8395 + 6.83551i −0.916164 + 0.528948i −0.882409 0.470482i \(-0.844080\pi\)
−0.0337550 + 0.999430i \(0.510747\pi\)
\(168\) 0 0
\(169\) −3.53425 + 6.12151i −0.271866 + 0.470885i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.1999 + 5.88891i 0.775484 + 0.447726i 0.834827 0.550512i \(-0.185567\pi\)
−0.0593437 + 0.998238i \(0.518901\pi\)
\(174\) 0 0
\(175\) 0.245424 + 6.71565i 0.0185523 + 0.507655i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −16.5727 −1.23870 −0.619350 0.785115i \(-0.712604\pi\)
−0.619350 + 0.785115i \(0.712604\pi\)
\(180\) 0 0
\(181\) 7.19552 + 12.4630i 0.534839 + 0.926368i 0.999171 + 0.0407069i \(0.0129610\pi\)
−0.464332 + 0.885661i \(0.653706\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −12.4567 + 0.227540i −0.915838 + 0.0167291i
\(186\) 0 0
\(187\) 6.18964 + 3.57359i 0.452631 + 0.261327i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.97170 0.432097 0.216049 0.976383i \(-0.430683\pi\)
0.216049 + 0.976383i \(0.430683\pi\)
\(192\) 0 0
\(193\) 14.1046 + 8.14331i 1.01527 + 0.586168i 0.912731 0.408560i \(-0.133969\pi\)
0.102542 + 0.994729i \(0.467302\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.5233i 0.820998i −0.911861 0.410499i \(-0.865355\pi\)
0.911861 0.410499i \(-0.134645\pi\)
\(198\) 0 0
\(199\) 4.79943 + 8.31285i 0.340222 + 0.589283i 0.984474 0.175531i \(-0.0561643\pi\)
−0.644251 + 0.764814i \(0.722831\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.41166 1.39237i 0.169265 0.0977253i
\(204\) 0 0
\(205\) 10.5794 + 5.85303i 0.738896 + 0.408794i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −15.2325 17.1127i −1.05366 1.18371i
\(210\) 0 0
\(211\) 7.28207 12.6129i 0.501318 0.868308i −0.498681 0.866786i \(-0.666182\pi\)
0.999999 0.00152265i \(-0.000484675\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.7739 + 24.8964i −0.939375 + 1.69792i
\(216\) 0 0
\(217\) 0.689764i 0.0468242i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.31182 0.222777
\(222\) 0 0
\(223\) −6.55816 3.78635i −0.439167 0.253553i 0.264077 0.964501i \(-0.414933\pi\)
−0.703244 + 0.710949i \(0.748266\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 6.86640i 0.455739i −0.973692 0.227869i \(-0.926824\pi\)
0.973692 0.227869i \(-0.0731759\pi\)
\(228\) 0 0
\(229\) −22.1011 −1.46048 −0.730240 0.683191i \(-0.760592\pi\)
−0.730240 + 0.683191i \(0.760592\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.57806 1.48844i −0.168894 0.0975111i 0.413170 0.910654i \(-0.364421\pi\)
−0.582064 + 0.813143i \(0.697755\pi\)
\(234\) 0 0
\(235\) 3.76562 + 6.25555i 0.245642 + 0.408067i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −9.71289 −0.628275 −0.314137 0.949378i \(-0.601715\pi\)
−0.314137 + 0.949378i \(0.601715\pi\)
\(240\) 0 0
\(241\) 9.34287 + 16.1823i 0.601827 + 1.04239i 0.992544 + 0.121884i \(0.0388937\pi\)
−0.390717 + 0.920511i \(0.627773\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −0.212097 11.6113i −0.0135504 0.741818i
\(246\) 0 0
\(247\) −10.0775 3.33803i −0.641216 0.212394i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.10091 3.63888i −0.132608 0.229684i 0.792073 0.610426i \(-0.209002\pi\)
−0.924681 + 0.380742i \(0.875668\pi\)
\(252\) 0 0
\(253\) −37.0814 + 21.4090i −2.33129 + 1.34597i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −24.3946 + 14.0842i −1.52169 + 0.878548i −0.522018 + 0.852934i \(0.674821\pi\)
−0.999672 + 0.0256140i \(0.991846\pi\)
\(258\) 0 0
\(259\) −7.48859 −0.465318
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.34563 1.35425i −0.144637 0.0835065i 0.425935 0.904754i \(-0.359945\pi\)
−0.570572 + 0.821247i \(0.693279\pi\)
\(264\) 0 0
\(265\) 13.5682 + 22.5398i 0.833486 + 1.38461i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.64101 + 9.77052i −0.343938 + 0.595719i −0.985160 0.171637i \(-0.945094\pi\)
0.641222 + 0.767356i \(0.278428\pi\)
\(270\) 0 0
\(271\) −3.16690 + 5.48523i −0.192375 + 0.333204i −0.946037 0.324059i \(-0.894952\pi\)
0.753662 + 0.657263i \(0.228286\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 26.2622 0.959753i 1.58367 0.0578753i
\(276\) 0 0
\(277\) 11.1435i 0.669549i −0.942298 0.334774i \(-0.891340\pi\)
0.942298 0.334774i \(-0.108660\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 12.9061 + 22.3541i 0.769916 + 1.33353i 0.937608 + 0.347694i \(0.113035\pi\)
−0.167693 + 0.985839i \(0.553632\pi\)
\(282\) 0 0
\(283\) −24.9942 14.4304i −1.48575 0.857799i −0.485883 0.874024i \(-0.661502\pi\)
−0.999868 + 0.0162249i \(0.994835\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.29360 + 3.63361i 0.371500 + 0.214485i
\(288\) 0 0
\(289\) −7.57543 13.1210i −0.445614 0.771826i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 22.7742i 1.33048i −0.746628 0.665242i \(-0.768328\pi\)
0.746628 0.665242i \(-0.231672\pi\)
\(294\) 0 0
\(295\) 0.0688637 + 0.0380988i 0.00400940 + 0.00221820i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −9.92035 + 17.1825i −0.573709 + 0.993692i
\(300\) 0 0
\(301\) −8.55098 + 14.8107i −0.492870 + 0.853676i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.98614 1.19559i 0.113726 0.0684592i
\(306\) 0 0
\(307\) 14.0275 + 8.09880i 0.800593 + 0.462223i 0.843679 0.536849i \(-0.180385\pi\)
−0.0430854 + 0.999071i \(0.513719\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 28.3483 1.60749 0.803743 0.594977i \(-0.202839\pi\)
0.803743 + 0.594977i \(0.202839\pi\)
\(312\) 0 0
\(313\) 2.67539 1.54464i 0.151222 0.0873081i −0.422480 0.906372i \(-0.638840\pi\)
0.573702 + 0.819064i \(0.305507\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −20.8236 + 12.0225i −1.16957 + 0.675251i −0.953579 0.301144i \(-0.902632\pi\)
−0.215991 + 0.976395i \(0.569298\pi\)
\(318\) 0 0
\(319\) −5.44500 9.43101i −0.304861 0.528035i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.19928 5.80476i −0.0667298 0.322986i
\(324\) 0 0
\(325\) 10.3165 6.46975i 0.572255 0.358877i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.19434 + 3.80071i 0.120978 + 0.209540i
\(330\) 0 0
\(331\) −20.7717 −1.14171 −0.570857 0.821049i \(-0.693389\pi\)
−0.570857 + 0.821049i \(0.693389\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −0.884231 1.46891i −0.0483107 0.0802550i
\(336\) 0 0
\(337\) −2.10139 1.21324i −0.114470 0.0660892i 0.441672 0.897177i \(-0.354386\pi\)
−0.556142 + 0.831087i \(0.687719\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.69739 0.146072
\(342\) 0 0
\(343\) 16.3885i 0.884896i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.48834 1.43664i −0.133581 0.0771230i 0.431720 0.902008i \(-0.357907\pi\)
−0.565301 + 0.824885i \(0.691240\pi\)
\(348\) 0 0
\(349\) 5.89385 0.315490 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.0238i 0.639962i −0.947424 0.319981i \(-0.896324\pi\)
0.947424 0.319981i \(-0.103676\pi\)
\(354\) 0 0
\(355\) 22.2444 + 12.3067i 1.18061 + 0.653172i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.26590 3.92466i 0.119590 0.207136i −0.800015 0.599979i \(-0.795175\pi\)
0.919605 + 0.392844i \(0.128509\pi\)
\(360\) 0 0
\(361\) −2.20143 + 18.8720i −0.115865 + 0.993265i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.33026 + 4.21195i −0.121971 + 0.220463i
\(366\) 0 0
\(367\) 29.9143 17.2710i 1.56151 0.901539i 0.564407 0.825496i \(-0.309105\pi\)
0.997105 0.0760429i \(-0.0242286\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 7.90659 + 13.6946i 0.410490 + 0.710989i
\(372\) 0 0
\(373\) 24.0801i 1.24682i −0.781894 0.623411i \(-0.785746\pi\)
0.781894 0.623411i \(-0.214254\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.37008 2.52307i −0.225071 0.129945i
\(378\) 0 0
\(379\) −30.1565 −1.54904 −0.774518 0.632552i \(-0.782008\pi\)
−0.774518 + 0.632552i \(0.782008\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −33.6192 19.4101i −1.71786 0.991809i −0.922797 0.385286i \(-0.874103\pi\)
−0.795066 0.606522i \(-0.792564\pi\)
\(384\) 0 0
\(385\) 15.7932 0.288486i 0.804898 0.0147026i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −18.6935 32.3781i −0.947799 1.64164i −0.750048 0.661383i \(-0.769970\pi\)
−0.197750 0.980252i \(-0.563364\pi\)
\(390\) 0 0
\(391\) −11.0779 −0.560236
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.3799 + 14.0414i 1.27700 + 0.706501i
\(396\) 0 0
\(397\) −13.4790 7.78211i −0.676492 0.390573i 0.122040 0.992525i \(-0.461056\pi\)
−0.798532 + 0.601952i \(0.794390\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.3113 19.5918i 0.564860 0.978366i −0.432203 0.901777i \(-0.642263\pi\)
0.997063 0.0765898i \(-0.0244032\pi\)
\(402\) 0 0
\(403\) 1.08244 0.624949i 0.0539203 0.0311309i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 29.2848i 1.45159i
\(408\) 0 0
\(409\) −18.1239 31.3915i −0.896169 1.55221i −0.832351 0.554249i \(-0.813005\pi\)
−0.0638187 0.997962i \(-0.520328\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.0409666 + 0.0236521i 0.00201583 + 0.00116384i
\(414\) 0 0
\(415\) 9.39671 0.171644i 0.461266 0.00842569i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.5598 −0.711293 −0.355647 0.934621i \(-0.615739\pi\)
−0.355647 + 0.934621i \(0.615739\pi\)
\(420\) 0 0
\(421\) −0.784161 + 1.35821i −0.0382177 + 0.0661950i −0.884502 0.466537i \(-0.845501\pi\)
0.846284 + 0.532732i \(0.178835\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 6.00846 + 3.18226i 0.291453 + 0.154362i
\(426\) 0 0
\(427\) 1.20673 0.696705i 0.0583977 0.0337159i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −12.7303 22.0495i −0.613197 1.06209i −0.990698 0.136080i \(-0.956550\pi\)
0.377500 0.926009i \(-0.376784\pi\)
\(432\) 0 0
\(433\) 14.6212 8.44155i 0.702650 0.405675i −0.105684 0.994400i \(-0.533703\pi\)
0.808334 + 0.588725i \(0.200370\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 33.7090 + 11.1656i 1.61252 + 0.534125i
\(438\) 0 0
\(439\) 9.93240 17.2034i 0.474048 0.821075i −0.525511 0.850787i \(-0.676126\pi\)
0.999558 + 0.0297121i \(0.00945905\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.46422 + 2.57742i −0.212101 + 0.122457i −0.602288 0.798279i \(-0.705744\pi\)
0.390186 + 0.920736i \(0.372411\pi\)
\(444\) 0 0
\(445\) 14.0013 8.42830i 0.663726 0.399540i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −33.2207 −1.56778 −0.783892 0.620897i \(-0.786768\pi\)
−0.783892 + 0.620897i \(0.786768\pi\)
\(450\) 0 0
\(451\) 14.2096 24.6117i 0.669103 1.15892i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 6.27088 3.77485i 0.293983 0.176968i
\(456\) 0 0
\(457\) 11.7126i 0.547894i 0.961745 + 0.273947i \(0.0883293\pi\)
−0.961745 + 0.273947i \(0.911671\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −3.68501 + 6.38263i −0.171628 + 0.297269i −0.938989 0.343947i \(-0.888236\pi\)
0.767361 + 0.641215i \(0.221569\pi\)
\(462\) 0 0
\(463\) 28.8020i 1.33854i 0.743019 + 0.669271i \(0.233393\pi\)
−0.743019 + 0.669271i \(0.766607\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 34.1251i 1.57912i −0.613673 0.789561i \(-0.710309\pi\)
0.613673 0.789561i \(-0.289691\pi\)
\(468\) 0 0
\(469\) −0.515268 0.892471i −0.0237929 0.0412105i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 57.9188 + 33.4394i 2.66311 + 1.53755i
\(474\) 0 0
\(475\) −15.0756 15.7393i −0.691717 0.722169i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.4130 24.9640i −0.658546 1.14064i −0.980992 0.194048i \(-0.937838\pi\)
0.322446 0.946588i \(-0.395495\pi\)
\(480\) 0 0
\(481\) 6.78491 + 11.7518i 0.309365 + 0.535836i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.902102 1.63055i 0.0409623 0.0740396i
\(486\) 0 0
\(487\) 16.5796i 0.751294i 0.926763 + 0.375647i \(0.122579\pi\)
−0.926763 + 0.375647i \(0.877421\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.94615 8.56698i 0.223217 0.386623i −0.732566 0.680696i \(-0.761678\pi\)
0.955783 + 0.294073i \(0.0950109\pi\)
\(492\) 0 0
\(493\) 2.81748i 0.126893i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.2331 + 7.64011i 0.593584 + 0.342706i
\(498\) 0 0
\(499\) −15.4949 + 26.8380i −0.693649 + 1.20144i 0.276985 + 0.960874i \(0.410665\pi\)
−0.970634 + 0.240561i \(0.922669\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −11.1406 + 6.43203i −0.496735 + 0.286790i −0.727364 0.686252i \(-0.759255\pi\)
0.230629 + 0.973042i \(0.425922\pi\)
\(504\) 0 0
\(505\) 28.3911 17.0905i 1.26339 0.760516i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.35312 + 12.7360i 0.325921 + 0.564512i 0.981698 0.190442i \(-0.0609922\pi\)
−0.655777 + 0.754955i \(0.727659\pi\)
\(510\) 0 0
\(511\) −1.44664 + 2.50566i −0.0639957 + 0.110844i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.0328 0.384195i 0.926817 0.0169296i
\(516\) 0 0
\(517\) 14.8630 8.58118i 0.653676 0.377400i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −5.35528 −0.234619 −0.117310 0.993095i \(-0.537427\pi\)
−0.117310 + 0.993095i \(0.537427\pi\)
\(522\) 0 0
\(523\) −13.8388 + 7.98981i −0.605127 + 0.349370i −0.771056 0.636768i \(-0.780271\pi\)
0.165929 + 0.986138i \(0.446938\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.604376 + 0.348937i 0.0263270 + 0.0151999i
\(528\) 0 0
\(529\) 21.6833 37.5566i 0.942753 1.63290i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 13.1687i 0.570400i
\(534\) 0 0
\(535\) −28.6459 + 0.523258i −1.23847 + 0.0226224i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −27.2972 −1.17577
\(540\) 0 0
\(541\) −17.9500 31.0904i −0.771732 1.33668i −0.936613 0.350366i \(-0.886057\pi\)
0.164881 0.986313i \(-0.447276\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −3.90095 2.15820i −0.167098 0.0924469i
\(546\) 0 0
\(547\) −22.6473 + 13.0754i −0.968327 + 0.559064i −0.898726 0.438511i \(-0.855506\pi\)
−0.0696011 + 0.997575i \(0.522173\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.83979 + 8.57329i −0.120979 + 0.365235i
\(552\) 0 0
\(553\) 15.0983 + 8.71704i 0.642047 + 0.370686i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.64444 2.10412i 0.154420 0.0891543i −0.420799 0.907154i \(-0.638250\pi\)
0.575219 + 0.818000i \(0.304917\pi\)
\(558\) 0 0
\(559\) 30.9899 1.31073
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 40.5225i 1.70782i 0.520422 + 0.853909i \(0.325775\pi\)
−0.520422 + 0.853909i \(0.674225\pi\)
\(564\) 0 0
\(565\) 18.6602 0.340855i 0.785039 0.0143399i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 23.9522 1.00413 0.502064 0.864831i \(-0.332574\pi\)
0.502064 + 0.864831i \(0.332574\pi\)
\(570\) 0 0
\(571\) −7.78949 −0.325980 −0.162990 0.986628i \(-0.552114\pi\)
−0.162990 + 0.986628i \(0.552114\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −34.5084 + 21.6411i −1.43910 + 0.902498i
\(576\) 0 0
\(577\) 34.1385i 1.42121i 0.703593 + 0.710603i \(0.251578\pi\)
−0.703593 + 0.710603i \(0.748422\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.64899 0.234360
\(582\) 0 0
\(583\) 53.5541 30.9195i 2.21798 1.28055i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −19.1737 11.0700i −0.791384 0.456906i 0.0490654 0.998796i \(-0.484376\pi\)
−0.840450 + 0.541890i \(0.817709\pi\)
\(588\) 0 0
\(589\) −1.48735 1.67094i −0.0612853 0.0688498i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 35.8131 20.6767i 1.47067 0.849089i 0.471208 0.882022i \(-0.343818\pi\)
0.999458 + 0.0329325i \(0.0104846\pi\)
\(594\) 0 0
\(595\) 3.57595 + 1.97839i 0.146600 + 0.0811061i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.8243 29.1406i −0.687423 1.19065i −0.972669 0.232197i \(-0.925409\pi\)
0.285246 0.958454i \(-0.407925\pi\)
\(600\) 0 0
\(601\) 38.4939 1.57020 0.785100 0.619369i \(-0.212611\pi\)
0.785100 + 0.619369i \(0.212611\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.678932 37.1683i −0.0276025 1.51111i
\(606\) 0 0
\(607\) 22.1827i 0.900367i −0.892936 0.450183i \(-0.851359\pi\)
0.892936 0.450183i \(-0.148641\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.97629 6.88714i 0.160864 0.278624i
\(612\) 0 0
\(613\) 4.13445 + 2.38703i 0.166989 + 0.0964111i 0.581165 0.813786i \(-0.302597\pi\)
−0.414176 + 0.910197i \(0.635930\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 14.0178 8.09317i 0.564335 0.325819i −0.190549 0.981678i \(-0.561027\pi\)
0.754883 + 0.655859i \(0.227693\pi\)
\(618\) 0 0
\(619\) −13.4892 −0.542176 −0.271088 0.962555i \(-0.587383\pi\)
−0.271088 + 0.962555i \(0.587383\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.50684 4.91143i 0.340819 0.196772i
\(624\) 0 0
\(625\) 24.9333 1.82482i 0.997332 0.0729926i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.78832 + 6.56156i −0.151050 + 0.261626i
\(630\) 0 0
\(631\) 13.2207 + 22.8989i 0.526308 + 0.911592i 0.999530 + 0.0306488i \(0.00975735\pi\)
−0.473222 + 0.880943i \(0.656909\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 23.7348 + 39.4288i 0.941886 + 1.56469i
\(636\) 0 0
\(637\) −10.9542 + 6.32441i −0.434021 + 0.250582i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.27817 7.41000i 0.168977 0.292677i −0.769083 0.639149i \(-0.779287\pi\)
0.938061 + 0.346471i \(0.112620\pi\)
\(642\) 0 0
\(643\) −24.2681 14.0112i −0.957042 0.552548i −0.0617804 0.998090i \(-0.519678\pi\)
−0.895261 + 0.445541i \(0.853011\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.57376i 0.376383i 0.982132 + 0.188192i \(0.0602626\pi\)
−0.982132 + 0.188192i \(0.939737\pi\)
\(648\) 0 0
\(649\) 0.0924936 0.160204i 0.00363069 0.00628854i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 16.4168i 0.642439i 0.947005 + 0.321219i \(0.104093\pi\)
−0.947005 + 0.321219i \(0.895907\pi\)
\(654\) 0 0
\(655\) −20.9963 11.6162i −0.820394 0.453882i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −7.08162 12.2657i −0.275861 0.477805i 0.694491 0.719501i \(-0.255629\pi\)
−0.970352 + 0.241697i \(0.922296\pi\)
\(660\) 0 0
\(661\) −18.5170 32.0724i −0.720229 1.24747i −0.960908 0.276868i \(-0.910704\pi\)
0.240679 0.970605i \(-0.422630\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −8.88717 9.62429i −0.344630 0.373214i
\(666\) 0 0
\(667\) 14.6178 + 8.43960i 0.566004 + 0.326783i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.72453 4.71903i −0.105179 0.182176i
\(672\) 0 0
\(673\) 42.3293i 1.63167i 0.578282 + 0.815837i \(0.303723\pi\)
−0.578282 + 0.815837i \(0.696277\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.9856i 0.537510i 0.963209 + 0.268755i \(0.0866121\pi\)
−0.963209 + 0.268755i \(0.913388\pi\)
\(678\) 0 0
\(679\) 0.560033 0.970005i 0.0214921 0.0372254i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11.6668i 0.446416i 0.974771 + 0.223208i \(0.0716529\pi\)
−0.974771 + 0.223208i \(0.928347\pi\)
\(684\) 0 0
\(685\) −19.8106 32.9099i −0.756925 1.25742i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 14.3273 24.8156i 0.545825 0.945397i
\(690\) 0 0
\(691\) −15.7886 −0.600627 −0.300313 0.953841i \(-0.597091\pi\)
−0.300313 + 0.953841i \(0.597091\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.0384 8.45062i 0.532506 0.320550i
\(696\) 0 0
\(697\) 6.36760 3.67633i 0.241190 0.139251i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −2.64450 + 4.58042i −0.0998816 + 0.173000i −0.911635 0.411000i \(-0.865180\pi\)
0.811754 + 0.584000i \(0.198513\pi\)
\(702\) 0 0
\(703\) 18.1409 16.1478i 0.684198 0.609025i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 17.2497 9.95913i 0.648743 0.374552i
\(708\) 0 0
\(709\) −12.2529 21.2226i −0.460166 0.797031i 0.538803 0.842432i \(-0.318877\pi\)
−0.998969 + 0.0454011i \(0.985543\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.62075 + 2.09044i −0.135598 + 0.0782875i
\(714\) 0 0
\(715\) −14.7619 24.5229i −0.552064 0.917104i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −22.4239 + 38.8393i −0.836269 + 1.44846i 0.0567236 + 0.998390i \(0.481935\pi\)
−0.892993 + 0.450071i \(0.851399\pi\)
\(720\) 0 0
\(721\) 12.6442 0.470896
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.50405 8.77660i −0.204415 0.325955i
\(726\) 0 0
\(727\) −28.1376 16.2453i −1.04357 0.602504i −0.122726 0.992441i \(-0.539164\pi\)
−0.920842 + 0.389937i \(0.872497\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.65152 + 14.9849i 0.319988 + 0.554235i
\(732\) 0 0
\(733\) 11.1969i 0.413568i 0.978387 + 0.206784i \(0.0662998\pi\)
−0.978387 + 0.206784i \(0.933700\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.49009 + 2.01501i −0.128559 + 0.0742237i
\(738\) 0 0
\(739\) 0.466361 0.807761i 0.0171554 0.0297140i −0.857320 0.514783i \(-0.827872\pi\)
0.874476 + 0.485069i \(0.161206\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 23.3370 + 13.4736i 0.856153 + 0.494300i 0.862722 0.505678i \(-0.168758\pi\)
−0.00656939 + 0.999978i \(0.502091\pi\)
\(744\) 0 0
\(745\) −23.9601 13.2559i −0.877829 0.485658i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −17.2210 −0.629240
\(750\) 0 0
\(751\) 2.33645 + 4.04686i 0.0852584 + 0.147672i 0.905501 0.424343i \(-0.139495\pi\)
−0.820243 + 0.572015i \(0.806162\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.471994 + 25.8394i 0.0171776 + 0.940392i
\(756\) 0 0
\(757\) 37.0902 + 21.4140i 1.34807 + 0.778306i 0.987975 0.154612i \(-0.0494126\pi\)
0.360090 + 0.932918i \(0.382746\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.7169 0.605987 0.302994 0.952993i \(-0.402014\pi\)
0.302994 + 0.952993i \(0.402014\pi\)
\(762\) 0 0
\(763\) −2.32065 1.33983i −0.0840131 0.0485050i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0.0857182i 0.00309511i
\(768\) 0 0
\(769\) −7.70852 13.3516i −0.277976 0.481469i 0.692905 0.721029i \(-0.256330\pi\)
−0.970882 + 0.239559i \(0.922997\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −28.7343 + 16.5897i −1.03350 + 0.596691i −0.917985 0.396615i \(-0.870185\pi\)
−0.115514 + 0.993306i \(0.536852\pi\)
\(774\) 0 0
\(775\) 2.56432 0.0937133i 0.0921132 0.00336628i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −23.0813 + 4.76868i −0.826975 + 0.170856i
\(780\) 0 0
\(781\) 29.8774 51.7491i 1.06910 1.85173i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.48139 8.10014i 0.159948 0.289106i
\(786\) 0 0
\(787\) 12.8318i 0.457405i −0.973496 0.228702i \(-0.926552\pi\)
0.973496 0.228702i \(-0.0734483\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 11.2179 0.398862
\(792\) 0 0
\(793\) −2.18667 1.26248i −0.0776511 0.0448319i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.28485i 0.0809336i 0.999181 + 0.0404668i \(0.0128845\pi\)
−0.999181 + 0.0404668i \(0.987115\pi\)
\(798\) 0 0
\(799\) 4.44028 0.157086
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 9.79861 + 5.65723i 0.345786 + 0.199639i
\(804\) 0 0
\(805\) −20.9759 + 12.6268i −0.739305 + 0.445036i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.3304 0.609305 0.304652 0.952464i \(-0.401460\pi\)
0.304652 + 0.952464i \(0.401460\pi\)
\(810\) 0 0
\(811\) −24.7926 42.9420i −0.870586 1.50790i −0.861392 0.507941i \(-0.830407\pi\)
−0.00919378 0.999958i \(-0.502927\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 21.0529 0.384562i 0.737452 0.0134706i
\(816\) 0 0
\(817\) −11.2221 54.3173i −0.392612 1.90032i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 11.1029 + 19.2308i 0.387495 + 0.671160i 0.992112 0.125356i \(-0.0400073\pi\)
−0.604617 + 0.796516i \(0.706674\pi\)
\(822\) 0 0
\(823\) −16.5674 + 9.56519i −0.577503 + 0.333422i −0.760141 0.649759i \(-0.774870\pi\)
0.182637 + 0.983180i \(0.441537\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −13.0330 + 7.52461i −0.453202 + 0.261656i −0.709182 0.705026i \(-0.750935\pi\)
0.255980 + 0.966682i \(0.417602\pi\)
\(828\) 0 0
\(829\) −3.62995 −0.126074 −0.0630368 0.998011i \(-0.520079\pi\)
−0.0630368 + 0.998011i \(0.520079\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.11621 3.53120i −0.211914 0.122349i
\(834\) 0 0
\(835\) 15.7656 + 26.1903i 0.545592 + 0.906351i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 9.47453 16.4104i 0.327097 0.566549i −0.654838 0.755770i \(-0.727263\pi\)
0.981935 + 0.189221i \(0.0605963\pi\)
\(840\) 0 0
\(841\) 12.3535 21.3969i 0.425984 0.737826i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 13.8302 + 7.65152i 0.475772 + 0.263220i
\(846\) 0 0
\(847\) 22.3443i 0.767761i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −22.6954 39.3095i −0.777987 1.34751i
\(852\) 0 0
\(853\) 37.0287 + 21.3785i 1.26784 + 0.731987i 0.974579 0.224045i \(-0.0719264\pi\)
0.293260 + 0.956033i \(0.405260\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.71973 + 1.57024i 0.0929043 + 0.0536383i 0.545732 0.837960i \(-0.316252\pi\)
−0.452828 + 0.891598i \(0.649585\pi\)
\(858\) 0 0
\(859\) 3.53437 + 6.12170i 0.120591 + 0.208870i 0.920001 0.391916i \(-0.128188\pi\)
−0.799410 + 0.600786i \(0.794854\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0.464328i 0.0158059i −0.999969 0.00790296i \(-0.997484\pi\)
0.999969 0.00790296i \(-0.00251562\pi\)
\(864\) 0 0
\(865\) 12.7493 23.0443i 0.433488 0.783531i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 34.0888 59.0435i 1.15638 2.00291i
\(870\) 0 0
\(871\) −0.933701 + 1.61722i −0.0316373 + 0.0547973i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 15.0041 0.822948i 0.507232 0.0278207i
\(876\) 0 0
\(877\) 0.802896 + 0.463552i 0.0271119 + 0.0156530i 0.513495 0.858093i \(-0.328351\pi\)
−0.486383 + 0.873746i \(0.661684\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −4.50850 −0.151895 −0.0759477 0.997112i \(-0.524198\pi\)
−0.0759477 + 0.997112i \(0.524198\pi\)
\(882\) 0 0
\(883\) 29.6605 17.1245i 0.998156 0.576286i 0.0904542 0.995901i \(-0.471168\pi\)
0.907702 + 0.419615i \(0.137835\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 24.7321 14.2791i 0.830423 0.479445i −0.0235747 0.999722i \(-0.507505\pi\)
0.853997 + 0.520277i \(0.174171\pi\)
\(888\) 0 0
\(889\) 13.8310 + 23.9560i 0.463876 + 0.803457i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −13.5113 4.47544i −0.452138 0.149765i
\(894\) 0 0
\(895\) 0.676798 + 37.0515i 0.0226229 + 1.23849i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.531667 0.920874i −0.0177321 0.0307129i
\(900\) 0 0
\(901\) 15.9991 0.533007
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 27.5696 16.5960i 0.916445 0.551668i
\(906\) 0 0
\(907\) 13.6098 + 7.85765i 0.451907 + 0.260909i 0.708635 0.705575i \(-0.249311\pi\)
−0.256728 + 0.966484i \(0.582644\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −20.7125 −0.686237 −0.343119 0.939292i \(-0.611483\pi\)
−0.343119 + 0.939292i \(0.611483\pi\)
\(912\) 0 0
\(913\) 22.0909i 0.731103i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −12.4906 7.21143i −0.412475 0.238142i
\(918\) 0 0
\(919\) −30.6628 −1.01147 −0.505737 0.862688i \(-0.668779\pi\)
−0.505737 + 0.862688i \(0.668779\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 27.6888i 0.911388i
\(924\) 0 0
\(925\) 1.01742 + 27.8402i 0.0334526 + 0.915380i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 7.65011 13.2504i 0.250992 0.434731i −0.712807 0.701360i \(-0.752577\pi\)
0.963799 + 0.266629i \(0.0859099\pi\)
\(930\) 0 0
\(931\) 15.0518 + 16.9097i 0.493303 + 0.554193i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 7.73669 13.9841i 0.253017 0.457329i
\(936\) 0 0
\(937\) 3.83862 2.21623i 0.125402 0.0724011i −0.435987 0.899953i \(-0.643601\pi\)
0.561389 + 0.827552i \(0.310267\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 17.3400 + 30.0338i 0.565268 + 0.979073i 0.997025 + 0.0770825i \(0.0245605\pi\)
−0.431757 + 0.901990i \(0.642106\pi\)
\(942\) 0 0
\(943\) 44.0490i 1.43443i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.04606 + 0.603945i 0.0339925 + 0.0196256i 0.516900 0.856046i \(-0.327086\pi\)
−0.482907 + 0.875671i \(0.660419\pi\)
\(948\) 0 0
\(949\) 5.24283 0.170189
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −51.3773 29.6627i −1.66427 0.960868i −0.970639 0.240540i \(-0.922676\pi\)
−0.693633 0.720329i \(-0.743991\pi\)
\(954\) 0 0
\(955\) −0.243873 13.3509i −0.00789156 0.432025i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.5443 19.9952i −0.372784 0.645680i
\(960\) 0 0
\(961\) −30.7366 −0.991504
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 17.6300 31.8662i 0.567528 1.02581i
\(966\) 0 0
\(967\) 39.4450 + 22.7736i 1.26847 + 0.732350i 0.974698 0.223527i \(-0.0717570\pi\)
0.293769 + 0.955876i \(0.405090\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0.738715 1.27949i 0.0237065 0.0410608i −0.853929 0.520390i \(-0.825787\pi\)
0.877635 + 0.479329i \(0.159120\pi\)
\(972\) 0 0
\(973\) 8.52937 4.92443i 0.273439 0.157870i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.8443i 1.49868i 0.662184 + 0.749341i \(0.269630\pi\)
−0.662184 + 0.749341i \(0.730370\pi\)
\(978\) 0 0
\(979\) −19.2066 33.2668i −0.613846 1.06321i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −17.4012 10.0466i −0.555011 0.320436i 0.196130 0.980578i \(-0.437163\pi\)
−0.751141 + 0.660142i \(0.770496\pi\)
\(984\) 0 0
\(985\) −25.7625 + 0.470589i −0.820861 + 0.0149942i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −103.660 −3.29621
\(990\) 0 0
\(991\) −8.95274 + 15.5066i −0.284393 + 0.492583i −0.972462 0.233062i \(-0.925125\pi\)
0.688069 + 0.725646i \(0.258459\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 18.3890 11.0695i 0.582971 0.350928i
\(996\) 0 0
\(997\) 1.71445 0.989838i 0.0542972 0.0313485i −0.472606 0.881274i \(-0.656686\pi\)
0.526903 + 0.849926i \(0.323353\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 3420.2.bj.c.1189.6 20
3.2 odd 2 380.2.r.a.49.8 yes 20
5.4 even 2 inner 3420.2.bj.c.1189.8 20
15.2 even 4 1900.2.i.g.201.8 20
15.8 even 4 1900.2.i.g.201.3 20
15.14 odd 2 380.2.r.a.49.3 20
19.7 even 3 inner 3420.2.bj.c.2629.8 20
57.26 odd 6 380.2.r.a.349.3 yes 20
95.64 even 6 inner 3420.2.bj.c.2629.6 20
285.83 even 12 1900.2.i.g.501.3 20
285.197 even 12 1900.2.i.g.501.8 20
285.254 odd 6 380.2.r.a.349.8 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
380.2.r.a.49.3 20 15.14 odd 2
380.2.r.a.49.8 yes 20 3.2 odd 2
380.2.r.a.349.3 yes 20 57.26 odd 6
380.2.r.a.349.8 yes 20 285.254 odd 6
1900.2.i.g.201.3 20 15.8 even 4
1900.2.i.g.201.8 20 15.2 even 4
1900.2.i.g.501.3 20 285.83 even 12
1900.2.i.g.501.8 20 285.197 even 12
3420.2.bj.c.1189.6 20 1.1 even 1 trivial
3420.2.bj.c.1189.8 20 5.4 even 2 inner
3420.2.bj.c.2629.6 20 95.64 even 6 inner
3420.2.bj.c.2629.8 20 19.7 even 3 inner